Min-Max payoffs in a two-player location game

We consider a two-player, sequential location game in d-dimensional Euclidean space with arbitrarily distributed consumer demand. The objective for each player is to select locations so as to maximize their market share—the mass of consumers in the vicinity of their chosen locations. At each stage, the two players (Leader and Follower) choose one location each from a feasible set in sequence. We first show that (i) if the feasible locations form a finite set in R, Leader (the first mover) must obtain at least a 1 d+1 fraction of the market share in equilibrium in the single-stage game, and there exist games in which Leader obtains no more than 1 d+1 ; (ii) in the original Hotelling game (uniformly distributed consumers on the unit interval), Leader obtains 1 2 even in the multiple stage game, using a strategy which is oblivious of Follower’s locations. Furthermore, we exhibit a strategy for Leader, such that even if she has no information about the number of moves, her payoff must equal at least half the payoff of the single-stage game.